The paradox highlights cases where, in performing an action, an omnipotent being would be limiting its abilities (therefore rendering it very firmly not omnipotent); conversely if it was unable to perform such an action, it would also not be omnipotent. The paradox represents a reductio ad absurdum, with the conclusion that a truly omnipotent being cannot exist.

The most classic example of the paradox, a Morton's fork, is the "Paradox of the Stone":

Can an omnipotent being create a stone so heavy that it cannot lift it?

If yes: the being's power is limited, because it cannot lift the stone.

If no: the being's power is limited, because it cannot create the stone.

Either way, the allegedly omnipotent being has proven not to be omnipotent due to the logical contradiction present in both possible answers.

The stone paradox can be substituted with similar examples. E.g. Could an omnipotent being create another being more powerful than itself? Could an omnipotent being destroy itself? Could he create a wall he cannot climb? Could he beat himself at arm-wrestling? And so on. The situation crops up numerous times in different wordings but all mean the same thing.

Some variations gives other useful consequences:

Can God create a cryptography/key exchange system so secure that he himself cannot crack/bypass?

Some have suggested that the concept of any paradox presents a challenge to the concept of omnipotence. For example, the thirteenth century theologian Thomas Aquinas pondered whether God could create a triangle (on a flat surface, no cheating with non-Euclidean geometry) in which the angles do not add up to 180 degrees. Or similarly, could God create a flat circle where π = 3? To do so would be to create a shape which cannot possibly exist in our universe - such a triangle can only exist on a curved surface and even warping space to do so is still placing the object on a curved surface. Yet if God cannot do so, then his powers must be limited by the laws of mathematics, and therefore he could not be fully omnipotent. This can be further complicated by asking "Can an omnipotent being create a triangle with four sides?" The issue here is that an object with four sides ceases to be a triangle and becomes a quadilateral - naturally, an omnipotent being could just rig up human language so that a four sided object was called "a triangle", or try to pass off a straight side as having a 180° corner in it. For the purposes of the paradox the aim is for the being to create a three sided object with four sides, without resorting to such semantic trickery.

Paradoxes do not necessarily have to ask for strange forces (like unmovable and unstoppable) and strange physics. Omnipotence can equally be limited by non-contradictory but very plain examples. For example consider an individual making the statement: "God cannot make me love him with my own free will". God could "miracle" some love in someone, but that wouldn't be an act of free will. God could torture someone until he "loved him", but similarly, that would be an act of force, morally indistinct from a criminal torturing someone in order to make them say something. God could argue that "he does not need X's love", however this would not remove the original problem that this represents a case where a mere mortal individual can limit God's omnipotence.

Most repudiations of the omnipotence paradox usually concern the definition of omnipotence, reducing it to merely an argument about semantics. These refutations assert that the paradox misrepresents omnipotence and the nature of God, as in their view, omnipotence is not necessarily bound by the laws of logic, physics or mathematics. Non-believers often have trouble grasping this idea, but those happy to accept that God is truly omnipotent regardless of what logic says have no trouble dealing with it. So, for example, when the empty mathematical logic of non-believers fails to apply the notion of absolute consistently to 'absolute power', 'absolute knowledge', and 'absolute love', the notion that omnipotence is not necessarily bound by logic takes on a certain attractiveness.

Another argument is that an omnipotent being does have every conceivable ability, including the ability to limit their own power (and hence stop being omnipotent). In this refutation, it is claimed that such a being is unlikely to make that move. Hence they could create a stone so heavy that they could not lift it; they just wouldn't.

The problem here is that the question then becomes "could the being complete the sequence of creating a stone too heavy for it to lift and then lifting it?"

If the paradox is reworded slightly to the common variant of "what happens when an unstoppable force meets an unmovable object?" then we can clearly spot a contradiction in the paradox itself, a similar problem to creating a four sided triangle as mentioned above. Namely, that if there is an unstoppable force, then there is no unmovable object that could be met by it, as if there was such an unmovable object, there could be no unstoppable force. The two are therefore mutually exclusive in existence, and it is a logical contradiction for both to exist at the same time. However, applying this contradiction to the God-based forms of the paradox is difficult, as these revolve around the omnipotent being's ability to create. And of course, the question is then raised; can an omnipotent being break this logical dichotomy of unstoppable and unmovable?

One solution of the omnipotence paradox is to make God omnipotent but still bound within the laws of logic. So while God could happily create matter out of absolutely nothing, violating conservation of energy, suddenly reverse the orbit of the planet Earth, violating conservation of momentum, or perhaps even make a Pot Noodle taste nice God would still be bound within the laws of logic. This is, of course, playing with the definition of omnipotence and it's generally up to the religion in question to determine the extent of the deity's omnipotence. This response if often said as something similar to, "God is able to do all that is able to be done." This approach could be interpreted as skewering itself on the horns of a metaphysical version of the Euthyphro dilemma - why is God the ultimate authority if there exist things even more fundamental?